Composite Functions and Their Domains

The domain of a composite function is the set of all possible input values for the outer function, considering the restrictions imposed by both the inner and outer functions.

To find the domain of a composite function, first determine the domain of the inner function (the function being applied first), then find the domain of the composed function (the function resulting from the composition). Finally, ensure that the final domain includes the restrictions from both the inner and outer function's domains.

The composite function is formed by applying one function to the output of another function.

Represented as (f o g)(x), it means f(g(x)) where the output of g(x) becomes the input for f(x).

The domain of a function is the set of all possible input values for which the function is defined.

The composite function (f o n)(x) = √(x + 5) has a domain of {x | x ≥ -5} because the restriction from the square root function (√x) requires x + 5 to be greater than or equal to zero.

The composite function (h o p)(x) = 1/(x² - 10x + 25) has a domain of {x | x ≠ 5}. This is because the outer function h(x) = 1/x is undefined when x = 0, and the expression x² - 10x + 25 equals 0 when x = 5.

The composite function (f o m)(x) = √(x²) = |x| has a domain of {x | x ∈ R} because the square root of x² is always a non-negative number, regardless of the input value of x.